It's 8.45am on March 23 in Rwanda, and an hour and a half into the journey north from the capital, Kigali. My driver, John, is looking increasingly worried. We're on a small, winding, rock-strewn road, and no one along it appears to have heard of our destination.

Our main problem is that our map - while the most recent, topographically detailed available - is a photocopy of a 1:100, 000 version printed by Rwanda's Belgian colonial government in 1937. Consequently it shows only a few roads and none of the modern towns - not that there are any nearby. It does mark contours, rivers, altitudes and bridges - jolly useful if you're exploring a river, as the three explorers who have sent it to me are doing.

But not much use, frankly, if you're haring around cloud-capped mountains trying to find them.

Fortunately, in the aptly named "Land of a Thousand Hills", the gods join us. As we turn the umpteenth bend, and look down at the rust-coloured river below, we spot three tiny white inflatable dinghies in the water. And as we get closer, three white men in filthy shorts and T-shirts stand out from the hundreds of Rwandans gaggled on the river banks. The locals have come to see with their own eyes the mzungus (white men) who could be about to rewrite history.

The three, Neil McGrigor, 44, Cam McLeay, 43, and Garth MacIntyre, 43, are on a mission which has been attempted in parts by some of Britain's finest explorers - Speke, Grant, Gordon, Baker, Livingstone - but never fully achieved: to follow the Nile all the way back to its "longest" source way above Lake Victoria. When I join them, they are about 4,100 miles from their starting point at the Nile Delta.

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If they reach their destination, the precise spot where the stream that becomes the Nile springs from the ground, which they think is about 95 miles away, they believe they will have done what no man has done before.

McGrigor, a Briton, and McLeay and MacIntyre, both New Zealanders, claim that they will have become the first explorers to travel the entire length of the Nile. They believe they will have proved that the source of the Rukarara river in the heart of the Nyungwe National Park in Rwanda is in fact also the source of the world's longest river: the tributary of the Nile furthest from its Mediterranean outlet.

They also think that by monitoring their route with the help of GPS and MarineTrack, and registering their data with the Royal Geographical Society, they will have proved that the Nile is far, far longer, and much more winding, than previously thought. They will have redrawn the map of Africa.

So far they have travelled for 70 days - in two phases. In November 2005, on Day 53, they were ambushed by Ugandan rebels who killed their driver and friend, Steve Willis, and left the rest of the party with serious injuries. Each of the explorers went home to recuperate - McGrigor to Hampshire, McLeay to Kampala and MacIntyre to Wellington - and they resumed the expedition on March 3.

The day I join them, Day 17 of their second phase, sees a neat double repetition of history. Not only was The Daily Telegraph a co-sponsor of Henry Morton Stanley's expedition to the Nile in the 1870s, but today McGrigor, McLeay and MacIntyre will pick up a hamper from Fortnum & Mason, which also sponsored and supplied Stanley with provisions such as thick-cut marmalade, humbugs and sardines.

As I scramble down the river bank (and, to the hilarity of the villagers, almost slip into their latrine pit), the men are fine-tuning their boats for some large rapids ahead. All three look fit and happy - and unbelievably muddy and wet. After spending a day with them, they promise, I'll look exactly the same - particularly if the next stretch of river is as testing as they expect.

Ideally, McGrigor says, we will ascend the rapids - which on the map look "difficult, but hopefully not impossible" - by Zapcat, a light-hulled catamaran version of a Rib (rigid inflatable boat). If we can't, another plan will have to be made: carrying the boats ourselves, or arranging porterage. Whatever happens, it will be a long day.

Not knowing what every day will bring gives the journey its edge, the men say. Above the Murchison Falls, for instance, the 60 miles of Fola rapids were so massive and impassable that the only way out was on a Fib (flying inflatable boat), which they crashed, just before the rebel attack.

Over the rapids from Lake Kyoga to Jinja, the only way out was to helicopter each craft to safer waters. In Tanzania and in the Akagera National Park in Rwanda, the bush was so thick that they had to hire porters to help carry the boats, and men on bicycles to move the engines.

Speeding off in his four-metre inflatable Zapcat, in hot pursuit of the New Zealanders, McGrigor admits that several times he has feared for their lives: when they were crossing Lake Victoria "the stormy skies were so black, the only way I could see the other guys behind me was when lightning struck"; having a gigantic Nile crocodile snap at him at the Fola rapids; taking off in the Fib on waters teeming with hippos; and being forced by Ugandan rebels to climb on to the roof of the Land Rover with a broken leg, and then watching them set it alight.

When McLeay first told McGrigor, in 2003, about his childhood dream of ascending the Nile, the British adventurer wasn't convinced it would be possible. But having undertaken the longest free abseil in the world (550ft) in an underground cave in Oman, set a Transatlantic record of 11 days 14 hours in a yacht in 1998 and broken the record for circum-navigating Britain in a vessel under 30ft long, McGrigor isn't a man to shirk a challenge. If anything, the logistics that the trip would entail enticed him.

"For me, the best part of a journey like this is planning it," he shouts, deftly avoiding floating trees, sandbanks and shallows as we whizz through the churning waters. He points out the extensive medical kit and the maps rolled up in waterproof vials, and talks me through the reconnaissance journeys he has made by helicopter to plan their route, the Google Earth journey he has mapped out on his lap top, the GPS and computerised tracking equipment he has installed to trace their journey, and the visas, customs forms and security clearance he has had to get from five African governments.

Making sure the three of them would be reasonably comfortable has also been a challenge. The four-metre boats were chosen to be as light as possible, while being strong enough to carry sufficient food, water, medical supplies and fuel to keep them going for long distances. The small size, though, meant that there was nowhere to sleep, hence the biminis - strong canopy roofs which provide shade in temperatures that often hit 50 degrees C - and a clean, dry base on which to erect a tent or put a sleeping bag.

If there's one thing they haven't been short of on the trip, it's sleep, McLeay tells me when we stop upstream after an hour, and start a two-hour hike on the river banks in order to ascertain the size of the waterfalls ahead. Bugs, he says, often came out in their thousands after dark, as did massive Nile crocodiles. "In Uganda, we'd look out with our torches and see their eyes around us in the water. So really the best place to be once the sun had gone down was on the roof, in a tent."

After 11 hours in his tent, each man was more than ready to get up at dawn. Not that there was much privacy, even that early, for morning ablutions. "In southern Sudan, where some people had never seen mzungus, crowds would watch us do everything - and I mean everything - and then afterwards would even inspect what we'd produced in the sand, " McGrigor says, grinning.

"It was like being some rare species in a zoo, being prodded and stared at." While people have generally been "more than friendly - incredibly helpful and welcoming wherever we've been" - it's the landscapes that will stay in his mind for ever. For MacIntyre, travelling through Sudan - "with its amazing coloured desert and beautiful light" - reminded him of the six months he'd spent in the Sahara with his wife, Sue, during a two-year expedition through Africa from 1988.